Some inequalities between \(K^ 3_ X\) and \(\chi({\mathcal O}_ X)\) for a threefold of general type (Q1358243)

On a smooth minimal threefold of general type, there are inequalities between \(K_X^3\) and \(\chi ({\mathcal O}_X)\) as we have on a minimal surface of general type, namely \[ -{7\over 6} K_X^3-1 \leq\chi ({\mathcal O}_X) \leq- {1\over 72} K_X^3. \] The right inequality comes from Miyaoka's inequality. The left inequality comes from the inequality between some multiple of \(K_X^3\) and the dimension of the cohomology \(H^0 (X, {\mathcal O}_X (nK_X))\). These inequalities give some restrictions on the existence of smooth minimal threefolds of general type. It means that there are inequalities between \(c_1 \cdot c_2\) and \(c^3_1\).